# Efficient projection and backprojection scheme for spherically symmetric basis functions in divergent beam geometry

## Abstract

In cone-beam transmission tomography the measurements are performed with a divergent beam of x-rays. The reconstruction with iterative methods is an approach that offers the possibility to reconstruct the corresponding images directly from these measurements. Another approach based on spherically symmetric basis functions (blobs) has been reported with results demonstrating a better image quality for iterative reconstruction algorithms. When combining the two approaches (i.e., using blobs in iterative cone-beam reconstruction of divergent rays) the problem of blob sampling without introducing aliasing must be addressed. One solution to this problem is to select a blob size large enough to ensure a sufficient sampling, but this prevents a high resolution reconstruction, which is not desired. Another solution is a heuristic low-pass filtering, which removes this aliasing, but neglects the different contributions of blobs to the absorption depending on the spatial position in the volume and, therefore, cannot achieve the best image quality. This article presents a model of sampling the blobs which is motivated by the beam geometry. It can be used for high resolution reconstruction and can be implemented efficiently.

- Authors:

- Philips Research Europe, Roentgenstrasse 24-26, 22315 Hamburg (Germany)

- Publication Date:

- OSTI Identifier:
- 20853833

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Medical Physics; Journal Volume: 33; Journal Issue: 12; Other Information: DOI: 10.1118/1.2388570; (c) 2006 American Association of Physicists in Medicine; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 62 RADIOLOGY AND NUCLEAR MEDICINE; ABSORPTION; ALGORITHMS; BEAMS; COMPUTERIZED TOMOGRAPHY; IMAGE PROCESSING; IMAGES; ITERATIVE METHODS; SAMPLING; SPATIAL RESOLUTION; X RADIATION

### Citation Formats

```
Ziegler, Andy, Koehler, Thomas, Nielsen, Tim, and Proksa, Roland.
```*Efficient projection and backprojection scheme for spherically symmetric basis functions in divergent beam geometry*. United States: N. p., 2006.
Web. doi:10.1118/1.2388570.

```
Ziegler, Andy, Koehler, Thomas, Nielsen, Tim, & Proksa, Roland.
```*Efficient projection and backprojection scheme for spherically symmetric basis functions in divergent beam geometry*. United States. doi:10.1118/1.2388570.

```
Ziegler, Andy, Koehler, Thomas, Nielsen, Tim, and Proksa, Roland. Fri .
"Efficient projection and backprojection scheme for spherically symmetric basis functions in divergent beam geometry". United States.
doi:10.1118/1.2388570.
```

```
@article{osti_20853833,
```

title = {Efficient projection and backprojection scheme for spherically symmetric basis functions in divergent beam geometry},

author = {Ziegler, Andy and Koehler, Thomas and Nielsen, Tim and Proksa, Roland},

abstractNote = {In cone-beam transmission tomography the measurements are performed with a divergent beam of x-rays. The reconstruction with iterative methods is an approach that offers the possibility to reconstruct the corresponding images directly from these measurements. Another approach based on spherically symmetric basis functions (blobs) has been reported with results demonstrating a better image quality for iterative reconstruction algorithms. When combining the two approaches (i.e., using blobs in iterative cone-beam reconstruction of divergent rays) the problem of blob sampling without introducing aliasing must be addressed. One solution to this problem is to select a blob size large enough to ensure a sufficient sampling, but this prevents a high resolution reconstruction, which is not desired. Another solution is a heuristic low-pass filtering, which removes this aliasing, but neglects the different contributions of blobs to the absorption depending on the spatial position in the volume and, therefore, cannot achieve the best image quality. This article presents a model of sampling the blobs which is motivated by the beam geometry. It can be used for high resolution reconstruction and can be implemented efficiently.},

doi = {10.1118/1.2388570},

journal = {Medical Physics},

number = 12,

volume = 33,

place = {United States},

year = {Fri Dec 15 00:00:00 EST 2006},

month = {Fri Dec 15 00:00:00 EST 2006}

}